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If two equal chords of a circle intersects within the circle,prove that the segments of one chord are equal to corresponding chords of the other chords?
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If two equal chords of a circle intersects within the circle,prove tha...
Introduction
When two equal chords intersect within a circle, we can prove that the segments of one chord are equal to the corresponding segments of the other chord. This relationship is rooted in the properties of circles and chords.
Consider the Setup
- Let the circle be centered at O.
- Let the two equal chords be AB and CD, intersecting at point P within the circle.
- Since AB and CD are equal chords, we have AB = CD.
Segments of Chords
- Let AP = x, PB = y, CP = z, and PD = w.
- By the property of intersecting chords, we know that:
- AP * PB = CP * PD
Setting Up the Equality
- Since AB equals CD, we can express them in terms of their segments:
- x + y = z + w
Using Equal Lengths
- Since the chords are equal, we can also say:
- x + y = z + w implies y = w and x = z.
Conclusion
- Therefore, we conclude that the segments of chord AB are equal to the corresponding segments of chord CD:
- AP = CP and PB = PD.
This proves that if two equal chords intersect within a circle, their segments are equal to the corresponding segments of the other chord.
When two equal chords intersect within a circle, we can prove that the segments of one chord are equal to the corresponding segments of the other chord. This relationship is rooted in the properties of circles and chords.
Consider the Setup
- Let the circle be centered at O.
- Let the two equal chords be AB and CD, intersecting at point P within the circle.
- Since AB and CD are equal chords, we have AB = CD.
Segments of Chords
- Let AP = x, PB = y, CP = z, and PD = w.
- By the property of intersecting chords, we know that:
- AP * PB = CP * PD
Setting Up the Equality
- Since AB equals CD, we can express them in terms of their segments:
- x + y = z + w
Using Equal Lengths
- Since the chords are equal, we can also say:
- x + y = z + w implies y = w and x = z.
Conclusion
- Therefore, we conclude that the segments of chord AB are equal to the corresponding segments of chord CD:
- AP = CP and PB = PD.
This proves that if two equal chords intersect within a circle, their segments are equal to the corresponding segments of the other chord.
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If two equal chords of a circle intersects within the circle,prove that the segments of one chord are equal to corresponding chords of the other chords?
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If two equal chords of a circle intersects within the circle,prove that the segments of one chord are equal to corresponding chords of the other chords? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about If two equal chords of a circle intersects within the circle,prove that the segments of one chord are equal to corresponding chords of the other chords? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If two equal chords of a circle intersects within the circle,prove that the segments of one chord are equal to corresponding chords of the other chords?.
If two equal chords of a circle intersects within the circle,prove that the segments of one chord are equal to corresponding chords of the other chords? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about If two equal chords of a circle intersects within the circle,prove that the segments of one chord are equal to corresponding chords of the other chords? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If two equal chords of a circle intersects within the circle,prove that the segments of one chord are equal to corresponding chords of the other chords?.
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